It seems to be relatively recent. The OED doesn’t list draw down specifically for this sense (the more usual sense means to invoke, to pull in, as in to draw down the wrath of God). Although there is this early 19th century cookery citation, meaning to boil down, to reduce:

1806 Culina 15 Put all those into a stew pan, with some water, and draw them down to a light brown colour.

The Big Dic does list a military sense meaning to withdraw or move troops for draw off, dating to the 17th century. Not quite the same sense as it doesn’t convey reduction in numbers though:

This has just sparked a memory of a bank assistant, when I got a mortgage a few years back, saying that now all the forms and legalities were in place he could ‘draw down’ the mortgage, meaning he could get the funds into the relevant account to make the purchase. It seemed like an odd phrase at the time but I assumed it was a specialist bank term - can’t seem to get quite the right sense from either of the oed entries.

I’ve heard it used with regard to lactation which is probably completely irrelevant. Is this sense in OED? Making milk available like bank funds? (This is my most tenuous yet, right up there with Totterdown!) No need to dignify this with a reply

This has just sparked a memory of a bank assistant, when I got a mortgage a few years back, saying that now all the forms and legalities were in place he could ‘draw down’ the mortgage, meaning he could get the funds into the relevant account to make the purchase.

A version of this was used just last night at a meeting I was at. Our organization is building a new building and the contractor said that his firm will be making their first “draw” this next week. Meaning a check is to be written to them from our funds.